Forcing randomized quads to stay coplanar?
As @batFINGER said, you can use dupli-face.

Create a plane and parent it to your displaced mesh (the one you shown in picture).



In the object properties of your displaced mesh, go to the duplication panel and select "Faces" (the blue button on the bottom part of the picture).

 

And.... that's all...

On the left, the "tri deformed original mesh", on the right "only planar quads"

(again, thanks to @batFINGER comment)